# Upper central series members are completely divisibility-closed in nilpotent group

## Contents

## Statement

Suppose is a nilpotent group. Then, the members of the upper central series of (with the possible exception of the zeroth member, the trivial subgroup) are completely divisibility-closed subgroups of . Explicitly, this means that if is a prime number such that is -divisible, then all the members of the upper central series of (possibly excepting the zeroth member, the trivial subgroup) are completely -divisibility-closed subgroups of .

In particular, the center and second center are both completely divisibility-closed subgroups, and hence also divisibility-closed subgroups.

## Related facts

### Similar facts

### Opposite facts

- Lower central series members need not be completely divisibility-closed in nilpotent group
- Center not is divisibility-closed

### Facts similar to proof technique

- Epicentral series members are completely divisibility-closed in nilpotent group
- Annihilator of divisibility-closed subgroup under bihomomorphism is completely divisibility-closed
- Complete divisibility-closedness is strongly intersection-closed

## Facts used

- Annihilator of divisibility-closed subgroup under bihomomorphism is completely divisibility-closed
- Complete divisibility-closedness is strongly intersection-closed
- Complete divisibility-closedness is transitive

## Proof

The proof of this is unusual in that the induction proceeds *downward* from the penultimate member (the largest). The idea is to shift the "take the root" operation between the elements in the iterated commutator map and note that the output is unaffected. This forces all roots of an element in the appropriate upper central series member to be in that upper central series member.